The convergence of the series  \(\sum \limits ^{\infty }_{i \,=\, 1}\,\overline{\,c_{\,i}}\,\left<\,f \,,\, f_{\,i} \,|\, a_{\,2} \,,\, \cdots \,,\, a_{\,n} \,\right>\)   for all  \(\left\lbrace \,c_{i}\,\right\rbrace _{i \,=\, 1}^{\,\infty } \,\in \, l^{\,2}\,(\,\mathbb {N}\,)\)   implies that  \(\left\lbrace \,\left<\,f \;,\; f_{\,i} \;|\; a_{\,2} \;,\; \cdots \;,\; a_{\,n} \,\right>\,\right\rbrace _{i \,=\, 1}^{\,\infty } \,\in \, l^{\,2}\,(\,\mathbb {N}\,)\) , ( see page 145 of [1]} ).
\(\text{Therefore,}\hspace{2.84544pt}\left<\,f \,,\, T_{F}\,\left\lbrace \,c_{i}\,\right\rbrace _{i \,=\, 1}^{\,\infty }\,\right>_{F} \,=\, \left<\;\left\lbrace \;\left<\,f \;,\; f_{\,i} \;|\; a_{\,2} \;,\; \cdots \;,\; a_{\,n} \,\right>\,\right\rbrace _{i \,=\, 1}^{\,\infty } \;,\; \left\lbrace \,c_{i}\,\right\rbrace _{i \,=\, 1}^{\,\infty }\;\right>_{l^{\,2}\,(\,\mathbb {N}\,)}\) 
